184 
DE. T. J. I’A. BEOMWICH ON THE 
aiid using this in equation (3 • 4) we deduce that 
(3-82) S._, (*) - (^ + j) S. W = I S„ (2) + . 
Combining (3'81) and (3’82) we have also 
(3 • 83) . S„_a {z) + S„+i {z) ^ ^ S„ (z). 
The relations (3-82), (3-83) hold equally for E,j (z) and C„ {z), as may be seen from (3-5) and (3-6). 
As E„ ( 2 ), S„ ( 2 ) are independent solutions of the equation (3‘8), it is evident that 
E„{z) ^ = const. 
dz dz 
Now when c is small, it is easy to verify from (3'4) and (3'G) that 
E„ ( 2 ) 
and accordingly we have 
(3-84) 
= 1. 
7t 
2/i+l 
+ O {z^), 
In the discussions of §6, when n, z are both large, it will be convenient to adopt 
the following notation :— 
(3’85) |E„( 2 )j = R, E„ ( 2 ) = Re^"^, so that S„ ( 2 ) = R sin i/r, C„ ( 2 ) = R cos 1 /^. 
Substituting from (3‘85) in (3'84) we deduce that 
(3-86) R2^ = l. 
dz 
Before leaving these preliminary formulm it will be convenient to quote the formula 
for in terms of our standard functions; namely 
(3-9) c'''" = 2 
71 = 0 K)' 
where 2 = r cos 0 = v/x and P„(/u) is Legendre’s polynomial of order n. 
This result follows at once from tlie formula given in Lamb’s ‘ Hydrodynamics,’ 
Art. 291, on using the relation (3’4) between i/r„(vr) and S„ already quoted. 
It is of course evident that an expansion of the type (3’9) might be anticipated, 
since each side satisfies the wave-equation, is symmetrical about the axis of 2 , and is 
continuous at r = 0 ; the determination of the numerical coefficients may be then 
carried out quickly by comparing the terms in on the two sides of the equation. 
§4. Plane Electromagnetic Waves Incident on a Spherical Obstacle. 
Suppose that the incident wave-train is travelling along the negative direction of 
the axis of 2 (that is, from 9 = U towards 0 = tt) ; and that it is polarized in the 
plane of yz (tliat is, in the plane 0 Suppose further that the electric force in 
